$BMO$, the space of functions of bounded mean oscillation, was first introduced by F. John and L. Nirenberg in 1961. It became a focus of attention when C. Fefferman proved that $BMO$ is the dual of the (real) Hardy space $H^1$ in 1971. In the past 30 years, this space was studied extensively by many mathematicians. With the help of $BMO$, many phenomena can be characterized clearly. In this review we discuss the connections between $BMO$ functions, the sharp function operator, Carleson measures, atomic decompositions and commutator operators in $\mathbf{R}^n$. We strive to cover some of the main developments in the theory, including $BMO$ in a bounded Lipschitz domain in $\mathbf{R}^n$ and in the product space $\mathbf{R} \times \mathbf{R}$.

Various irrationality results on certain infinite series and products representing q-functions were established in recent years by Borwein, Lubinsky, and Zhou [1], [7-10]. In all these papers, Pad´e approximants to appropriate functions were constructed to produce rational approximations that are too rapid to be consistent with rationality. The main purpose of this note is to show how an old and seemingly forgotten irrationality criterion of the present author [3], particularly suited for q-functions, can be used to deduce very easily much more general results.

A finite group $G$ is called $(l,m,n)$-generated, if it is a quotient group of the triangle group $T(l,m,n) = \langle x,y,z | x^l = y^m = z^n = xyz = 1 \rangle$.

In [16], the question of finding all triples $(p,q,r)$ such that non-abelian finite simple group $G$ is $(p,q,r)-$generated was posed. In this paper we partially answer this question for the sporadic group $HN$. In fact, we prove that the sporadic group $HN$ is $(p,q,r)-$generated if and only if $(p,q,r) \ne (2,3,5)$, where $p, q$ and $r$ are prime divisors of $|HN|$ and $p \lt q \lt r$.

C. H. Gibbons, M. R. S. Kulenovic and G. Ladas [1] have posed the following problem: Is there a solution of the difference equation: $$ x_{n+1} = \frac{\beta x_{n-1}}{\beta + x_n}, \quad x_{-1}, x_0 \gt 0, \beta \gt 0 \quad (n = 0, 1, 2, \dots) $$ such that $\displaystyle \lim_{n \to \infty} x_n = 0$? S. Stevic [2] gives an affirmative answer to this open problem and generalize this result to the equation of the form: $$ x_{n+1} = \frac{x_{n-1}}{g(x_n)}, \quad x_{-1}, x_0 \gt 0 \quad (n = 0, 1, 2, \dots) $$ by using his ingenious device. In this note, we generalize the result of Stevic to the equation of the form: $$ x_{n+1} = f(x_{n-1}, x_n), \quad x_{-1}, x_0 \gt 0 \quad (n = 0, 1, 2, \dots). $$ However our proof is simple and short.

In this paper, by using the section mappings, we introduce the abstract strong gliding hump property and the 0-gliding hump property in dual pair $\lt E,F \gt$, and show that the gliding hump properties can substitute the $AK$-property of the dual spaces for the characterizions of the barrelledness of normed spaces.

We construct an irreducible ergodic Harris chain $\{ X_n \}$ from a diffusion $\{ S_t \}$ and barriers $\rho^{\pm}(x)$. We show that $\{ X_n \}$ is exponentially uniformly ergodic in the sense of the operator norm under the Banach space $C_{\beta}$, where $\beta \in (0,1)$. Moreover, the sizes of the convergent rates $\alpha_{X}(\beta)$ and $\alpha_{S}(\beta)$ measured by the operator norm are studied. We give an upper bound of $\alpha_{X}(\beta)$ in terms of $\rho^{\pm}(x)$. The Ornstein-Uhlenbeck process and proper $\rho^{\pm}(x)$ are taken to show $\alpha_{X}(\beta) \lt \alpha_{S}(\beta)$ for $0 \lt \beta \lt 0.5$.

In this paper, the weighted boundedness of multilinear operators related to some non-convolution operators for the extreme cases are obtained. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Ülger, Saksman and Tylli have shown that if $X$ is a reflexive Banach space and $\mathcal{A}$ is a subalgebra of $K(X)$ such that $\mathcal{A}^*$ has the Schur property, then $\mathcal{A}$ is completely continuous. Here by introducing the concept of a strongly completely continuous subspace of an operator ideal, we improve their results. In particular, when $X$ is an $l_p$- direct sum and $Y$ is an $l_q$- direct sum of finite-dimensional Banach spaces with $1 \lt p \leq q \lt \infty$, we give a characterization of Schur property of the dual $\mathcal{M}^*$ of a closed subspace $\mathcal{M} \subseteq K(X,Y)$ in terms of strong complete continuity of $\mathcal{M}$.

In the present paper we first establish inverse versions of some new inequalities similar to Hilbert’s inequality involving series of nonnegative terms. Then, the integral analogues of our main results are also given. Our Theorems provide new estimates on these types of inequalities.

A class of hereditarily $\ell_{p}$ ($1 \leq p \lt \infty$) Banach sequence spaces is constructed and denoted by $X_{\alpha,p}$. Any constructed space is a dual space. We show that (i) the predual of any member $X$ of the class of $X_{\alpha,1}$ contains asymptotically isometric copies of $c_0$.(ii) Every infinite dimensional subspace of $X$ contains asymptotically isometric complemented copies of $\ell_{1}$, and consequently, the dual X$^*$ of X contains subspaces isometrically isomorphic to $C[0,1]^*$. (iii) Every member of the class of $X_{\alpha,p}$ ($1 \leq p \lt \infty$) fails the Dunford-Pettis property. (iv) We observe that all $X_{\alpha,p}$ spaces are Banach spaces without unconditional basis but all constructed spaces contain a subspace which is weakly sequentially complete with an unconditional basis which is weakly null sequence but not in norm. (v) All spaces have asymptotic-norming and Kadec-Klee property. The predual of any $X_{\alpha,p}$ is an Asplund space.

Easily verifiable sufficient criteria are established for the existence of periodic solutions of two mutualism systems with impulse. The approach is based on the coincidence degree and its related continuation theorem.

A class of distribution cosine functions is introduced and systematically analyzed. Connections with (local) integrated cosine functions as well as some characterizations and related examples are given.

In this paper we gain some results on the regularity and also the blow-up rates and constants of solutions to the equation $u'' - u^{p} = 0$ under some different situations. The blow-up rate and blow-up constant of $u^{(2n)}$ are $(p-2n+2)$ and $(\pm) (p-2n+2) \cdot \Pi_{i=0}^{n-1} (p-2i+2) (p-2i+1) E(0)^{p/2}$ respectively; blow-up rate and blow-up constant of $u^{(2n+1)}$ are $(p-2n+1)$ and $(p-2n+2) \Pi_{i=0}^{n-1} (p-2i+2) \cdot (p-2i+1) E(0)^{p-n}$ respectively, where $E(0) = u'(0)^{2} - \frac{2}{p+1} u(0)^{p+1}$.

In this paper, we first obtain the generalizations of the almost fixed point theorems on the almost $G$-convex sets and the Himmelberg fixed point theorems on a locally G-convex space. Next, we invoke non-convexity of constraint regions in place of convexity and we obtain the new fixed point theorems, "Let $X$ be an almost $G$-convex subset of a locally $G$-convex space $E$. If $T \in \Gamma^* - KKM(X,X)$ is compact and closed, then $T$ has a fixed point."

The main purpose of this paper is to study the thermal balance equations for the gas and solid interphase heat-transfer for the fast-igniting catalytic converter of automobiles: $$ \left\{ \begin{array}{ll} \frac{\partial}{\partial t} u(t,x) = -\alpha \frac{\partial}{\partial x} u(t,x) + av(t,x) - au(t,x) & \text{ for } t \gt 0, 0 \lt x \lt l; \\ \frac{\partial}{\partial t} v(t,x) = bu(t,x) - bv(t,x) + \lambda \exp(v(t,x)) & \text{ for } t \gt 0, 0 \lt x \lt l; \\ u(t,0) = \eta & \text{ for } t \geq 0; \\ u(0,x) = u_{0}(x) \text{ and } v(0,x) = v_{0}(x) & \text{ for } x \gt 0. \end{array} \right. $$ where $u_{0}$, $v_{0}$ are continuous functions on $[0,l]$ with $u_{0}(0) = \eta$. We establish some results concerning the existence and uniqueness of the mild solutions and classical solutions of the above differential system. The asymptotical behavior of the solution is also addressed.